Epistemic systems and Flagg and Friedman's translation

In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of the faithfulness of G\"{o}del (or Rasiowa-Sikorski) translation $(\cdot)^\Box$ of Heyting arithmetic $\bf HA$ to Shapiro's epistemic arithmetic $\bf EA$. In \S 2, we shall prove the faithfulness of $(\cdot)^\Box$ without using stability, by introducing another translation from an epistemic system to corresponding intuitionistic system which we shall call \it the modified Rasiowa-Sikorski translation\rm . That is, this introduction of the new translation simplifies the original Flagg and Friedman's proof. In \S 3, we shall give some applications of the modified one for the disjunction property ($\mathsf{DP}$) and the numerical existence property ($\mathsf{NEP}$) of Heyting arithmetic. In \S 4, we shall show that epistemic Markov's rule $\mathsf{EMR}$ in $\bf EA$ is proved via $\bf HA$. So $\bf EA$ $\vdash \mathsf{EMR}$ and $\bf HA$ $\vdash \mathsf{MR}$ are equivalent. In \S 5, we shall give some relations among the translations treated in the previous sections. In \S 6, we shall give an alternative proof of Glivenko's theorem. In \S 7, we shall propose several(modal-)epistemic versions of Markov's rule for Horsten's modal-epistemic arithmetic $\bf MEA$. And, as in \S 4, we shall study some meta-implications among those versions of Markov's rules in $\bf MEA$ and one in $\bf HA$. Friedman and Sheard gave a modal analogue $\mathsf{FS}$ (i.e. Theorem in \cite{fs}) of Friedman's theorem $\mathsf{F}$ (i.e. Theorem 1 in \cite {friedman}): \it Any recursively enumerable extension of $\bf HA$ which has $\mathsf{DP}$ also has $\mathsf{NPE}$\rm . In \S 8, we shall give a proof of our \it Fundamental Conjecture \rm $\mathsf{FC}$ proposed in Inou\'{e} \cite{ino90a} as follows: $\mathsf{FC}: \enspace \mathsf{FS} \enspace \Longrightarrow \enspace \mathsf{F}.$ This is a new type of proofs. In \S 9, I shall give discussions.Comment: 33 page

On Bergman's Diamond Lemma for Ring Theory

This expository paper deals with the Diamond Lemma for ring theory, which is proved in the first section of G.M. Bergman, The Diamond Lemma for Ring Theory, Advances in Mathematics, 29 (1978), pp. 178--218. No originality of the present note is claimed on the part of the author, except for some suggestions and figures. Throughout this paper, I shall mostly use Bergman's expressions in his paper.Comment: 15 page

Intuitionistic Propositional Calculus in the Extended Framework with Modal Operator. Part II

This paper is a continuation of Inou´e [5]. As already mentioned in the paper, a number of intuitionistic provable formulas are given with a Hilbert-style proof. For that, we make use of a family of intuitionistic deduction theorems, which are also presented in this paper by means of Mizar system [2], [1]. Our axiom system of intuitionistic propositional logic IPC is based on the propositional subsystem of H1-IQC in Troelstra and van Dalen [6, p. 68]. We also owe Heyting [4] and van Dalen [7]. Our treatment of a set-theoretic intuitionistic deduction theorem is due to Agata Darmochwał’s Mizar article “Calculus of Quantifiers. Deduction Theorem” [3].Takao Inoué - Department of Medical Molecular Informatics, Meiji Pharmaceutical University, Tokyo, Japan; Graduate School of Science and Engineering, Hosei University, Tokyo, Japan; Department of Applied Informatics, Faculty of Science and Engineering, Hosei University, Tokyo, JapanRiku Hanaoka - Keyaki-Sou 403, Midori-cho 5-17-27, Koganei-city, 184-0003, Tokyo, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-817.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Agata Darmochwał. Calculus of quantifiers. Deduction theorem. Formalized Mathematics, 2(2):309–312, 1991.Arend Heyting. Intuitionism. An introduction. Elsevier, Amsterdam, 3rd revised ed., 1971.Takao Inoué. Intuitionistic propositional calculus in the extended framework with modal operator. Part I. Formalized Mathematics, 11(3):259–266, 2003.Anne Sjerp Troelstra and Dirk van Dalen. Constructivism in mathematics. An introduction. Volume I, volume 121 of Studies in Logic and the Foundations of Mathematics. Amsterdam etc.: North-Holland, 1988. ISBN 0-444-70506-6.Dirk van Dalen. Logic and Structure. London: Springer, 2013. ISBN 978-1-4471-4557-8; 978-1-4471-4558-5. doi:10.1007/978-1-4471-4558-5.30111

A Scanning Electron Microscopic Study of the Basal Surface of the Corneal Endothelium and the Stromal and Endothelial Surfaces of Descemet's Membrane in Rats

The basal surface of the corneal endothelium and the stromal and endothelial surfaces of Descemet's membrane in rats were studied by scanning electron microscopy. We compared the fine structures of the two surfaces of Descemet's membrane both after sputter-coating with platinum and without sputter-coating. Fine structures were made clearly visible without metal coating by heating specimens to 300°C during observation. After sputter-coating, both surfaces of Descemet's membrane were composed of granular substances, but their sizes were larger on the stromal side than on the endothelial side. Both surfaces of Descemet's membrane observed without sputter-coating were composed of fine fibrous structures showing a felt-like appearance, but their diameters were thicker on the stromal surface. These results may reflect a difference in collagen types between the two surfaces of Descemet's membrane

Partial Differentiation of Real Ternary Functions

In this article, we shall extend the result of [19] to discuss partial differentiation of real ternary functions (refer to [8] and [16] for partial differentiation).Inoué Takao - Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanXie Bing - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Xiquan Liang, Piqing Zhao, and Ou Bai. Vector functions and their differentiation formulas in 3-dimensional Euclidean spaces. Formalized Mathematics, 18(1):1-10, 2010, doi: 10.2478/v10037-010-0001-2.Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Bing Xie, Xiquan Liang, and Hongwei Li. Partial differentiation of real binary functions. Formalized Mathematics, 16(4):333-338, 2008, doi:10.2478/v10037-008-0041-z

Partial Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

In this article, we define and develop partial differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [19] and [20]).Inoué Takao - Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanNaumowicz Adam - Institute of Computer Science, University of Białystok, Akademicka 2, 15-267 Białystok, PolandEndou Noboru - Gifu National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces n. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Takao Inoué, Noboru Endou, and Yasunari Shidama. Differentiation of vector-valued functions on n-dimensional real normed linear spaces. Formalized Mathematics, 18(4):207-212, 2010, doi: 10.2478/v10037-010-0025-7.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.Yatsuka Nakamura, Artur Korniłowicz, Nagato Oya, and Yasunari Shidama. The real vector spaces of finite sequences are finite dimensional. Formalized Mathematics, 17(1):1-9, 2009, doi:10.2478/v10037-009-0001-2.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Laurent Schwartz. Cours d'analyse. Hermann, 1981. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki, Yoshinori Fujisawa, and Yatsuka Nakamura. On replace function and swap function for finite sequences. Formalized Mathematics, 9(3):471-474, 2001

The Differentiable Functions from R into Rⁿ

In control engineering, differentiable partial functions from R into Rⁿ play a very important role. In this article, we formalized basic properties of such functions.Narita Keiko - Hirosaki-city, Aomori, JapanKorniłowicz Artur - Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Takao Inoué, Adam Naumowicz, Noboru Endou, and Yasunari Shidama. Partial differentiation of vector-valued functions on n-dimensional real normed linear spaces. Formalized Mathematics, 19(1):1-9, 2011, doi: 10.2478/v10037-011-0001-x.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Keiko Narita, Artur Kornilowicz, and Yasunari Shidama. More on the continuity of real functions. Formalized Mathematics, 19(4):233-239, 2011, doi: 10.2478/v10037-011-0032-3.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Hiroyuki Okazaki, Noboru Endou, Keiko Narita, and Yasunari Shidama. Differentiable functions into real normed spaces. Formalized Mathematics, 19(2):69-72, e2011, doi: 10.2478/v10037-011-0012-7.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011, doi: 10.2478/v10037-011-0008-3.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

Second-Order Partial Differentiation of Real Ternary Functions

In this article, we shall extend the result of [17] to discuss second-order partial differentiation of real ternary functions (refer to [7] and [14] for partial differentiation).Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Bing Xie, Xiquan Liang, and Hongwei Li. Partial differentiation of real binary functions. Formalized Mathematics, 16(4):333-338, 2008, doi:10.2478/v10037-008-0041-z.Bing Xie, Xiquan Liang, and Xiuzhuan Shen. Second-order partial differentiation of real binary functions. Formalized Mathematics, 17(2):79-87, 2009, doi: 10.2478/v10037-009-0009-7

Partial Differentiation, Differentiation and Continuity on n-Dimensional Real Normed Linear Spaces

In this article, we aim to prove the characterization of differentiation by means of partial differentiation for vector-valued functions on n-dimensional real normed linear spaces (refer to [15] and [16]).Inoué Takao - Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanNaumowicz Adam - Institute of Computer Science, University of Białystok, Akademicka 2, 15-267 Białystok, PolandEndou Noboru - Nagano National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Laurent Schwartz. Cours d'analyse. Hermann, 1981. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990